Most approaches discussed above compute the collision probability along a path by summing or multiplying the probabilities along different waypoints in the path. Boole's inequality is used to decouple the total probability in terms of individual waypoint probabilities. Such approaches tend to be overly conservative and rather than computing bounded collision probabilities along a path, the bound should be checked for each configuration along a path. Moreover, in most approaches, the collision probability computed along each waypoint is an approximation of the true value. On the one hand, such approximations can overly penalize paths and could gauge all plans to be infeasible. On the other hand some approximations can be lowerFor example, the approach in [1]} computes a value lower than the actual when the robot state covariance is small. than the true collision probability values and can lead to synthesizing unsafe plans.
